First, apply the Scale and Rotation to all objects (Ctrl+A in Object mode). It's important for the physics simulation. Applying the Location is also a good idea for rigid body simulations. Except here for the weight object which goes crazy if you do so. Applying the Location sets the object's origin to the world origin. For the weight object, it's better to set the origin to the center of mass (menu: Object > Set Origin > Set Origin to Center of Mass (Volume)).
Then maybe lower the mass of the weight object a bit. It's too heavy for the size. Your trebuchet is a mini version, about 1 meter tall :-)
Cloth + Rigid Body Collision
With the default settings, the Collision and Particle modifiers of a Rigid Body body doesn't follow the movement of the rigid object. In consequence, the Cloth object collides with the rigid body where it was at the start of the simulation:
To fix this you need to set Rigid Body > Collisions > Source to
Base. Then the collision follows the rigid body and the collision works as expected. In the example above, the rope (cloth) no longer passes the beam (rigid body). This works also for particles on the rigid body. (Related solved bug: T78859)
Firing a Stone
Attaching the Stone
The stone (just a simple mesh) can be attached to the rope (cloth) with a Vertex Parent relationship. Select exactly 1 or 3 vertices of the cloth in Edit mode and the stone object. Then parent them with the menu entry: Vertex > Make Vertex Parent.
Detaching the Stone
This is more difficult because you can't keyframe the parent relationship and remove it. But you can do a magician trick and fake it.
Just duplicate the stone, and place it where it will be detached. Keyframe the visibility of the first and the second stone. (Object Properties > Visibility > Show In >
[X] Renders and
[X] Viewports) The first stone is visible from frame 1 to 33 and the second stone is hidden and visible at frame 33.
In this example, the stones are visible so you see them both at the same time until frame 33:
The second stone can be a Rigid Body and be animated while it is hidden to gain momentum. Rigid Body > Settings >
[X] Animated must be checked to keyframe its location else it will fall down. In the example above, the location was just keyframed, with no rigid body physics.